Now in order to complete our analysis we must know what Moe would obtainfor the forces. The force is supposed to act along some line, and by the forcein the x -direction we mean the part of the total which is in the x -direction, which is the magnitude of the force times this cosine of itsangle with the x -axis. Now we see that Moe would use exactly the same projection as Joewould use, so we have a set of equations
Fx′=Fx, Fy′=Fy, Fz′=Fz. (11.3)
These would be the relationships between quantities as seen by Joe andMoe.
现在，为了完成我们的分析，我们应该知道，Moe能为力得到什么。力被假定是沿着某条线起作用，通过x方向的力，我们是指总的力在x方向的分量，它是力的大小，乘以，力与x轴的夹角的cosine。现在，我们看到，Moe使用的投影与Joe使用的投影一样，于是，我们就有一组方程：
Fx′=Fx, Fy′=Fy, Fz′=Fz. (11.3)
这就是Joe 和 Moe所看到的量之间的关系。

Fig. 11–1.Two parallel coordinate systems. 图11-1 两个平行的坐标系统。

The question is, if Joe knows Newton’slaws, and if Moe tries to write down Newton’s laws, will they also be correctfor him? Does it make any difference from which origin we measure the points?In other words, assuming that equations (11.1)are true, and the Eqs. (11.2)and (11.3)give the relationship of the measurements, is it or is it not true that
(a) m(d2x′/dt2)= Fx′
(b) m(d2y′/dt2)= Fy′ (11.4)
(c) m(d2z′/dt2)= Fz′
现在的问题是，如果Joe知道牛顿规律，并且，如果Moe想写出牛顿规律，那么，这些规律对他来说，也会是正确的吗？从不同原点出发，测量这些点，会有何不同？换句话说，假设方程(11.1)为真，方程（11.2）和（11.3）给出了测量的关系，那么，下面方程是否为真呢？
(a) m(d2x′/dt2)= Fx′
(b) m(d2y′/dt2)= Fy′ (11.4)
(c) m(d2z′/dt2)= Fz′
In order to test these equations we shall differentiate the formulafor x′ twice. First of all
dx′/dt=d(x−a)/dt=dx/dt−da/dt.
Now we shall assume that Moe’s origin is fixed (not moving) relativeto Joe’s; therefore a is a constant and da/dt=0 , so we find that
dx′/dt=dx/dt
and therefore
d2x′/dt2=d2x/dt2;
therefore we know that Eq. (11.4a)becomes
m(d2x/dt2)=Fx′.
(We also suppose that the masses measured by Joe and Moe are equal.)Thus the acceleration times the mass is the same as the other fellow’s. We havealso found the formula for Fx′ , for, substituting from Eq. (11.1),we find that
Fx′=Fx.
为了验证这些方程，我们将让方程对x′求微分两次。首先：
dx′/dt=d(x−a)/dt=dx/dt−da/dt.
现在，我们将假定Moe的原点，相对于Joe的原点，是固定的；因此，a是一个常数，且 da/dt=0，于是我们发现：
dx′/dt=dx/dt
从而：
d2x′/dt2=d2x/dt2;
因此，我们知道方程 (11.4a)就变为：
m(d2x/dt2)=Fx′.
(我们同样假设，Joe和Moe所测的质量，是相等的。)，这样，加速度乘以质量，就与另一伙计的一样。我们也为Fx′找到了公式，因为，为了替代公式（11.1），我们发现：
Fx′=Fx.
Therefore the laws as seen by Moe appear the same; he can writeNewton’s laws too, with different coordinates, and they will still be right.That means that there is no unique way to define the origin of the world,because the laws will appear the same, from whatever position they areobserved.
因此，Moe所看到的规律，是一样的；用不同的坐标，他也可以得出牛顿规律，且也是正确的。这就意味着，定义这个世界的原点，并非只有唯一的方法，因为，无论从哪个位置来观察规律，它们看上去都一样。
This is also true: if there is a piece ofequipment in one place with a certain kind of machinery in it, the same equipmentin another place will behave in the same way. Why? Because one machine, whenanalyzed by Moe, has exactly the same equations as the other one, analyzed byJoe. Since the equations are the same, the phenomena appear thesame. So the proof that an apparatus in a new position behaves the same as itdid in the old position is the same as the proof that the equations whendisplaced in space reproduce themselves. Therefore we say that the laws ofphysics are symmetrical for translational displacements, symmetrical in thesense that the laws do not change when we make a translation of our coordinates.Of course it is quite obvious intuitively that this is true, but it is interestingand entertaining to discuss the mathematics of it.
下面也是真的：如果某地有一个仪器，它有一定的机制，那么，在另外一个地方的同样的设备，将会有同样的表现。为什么？因为，一台机器，被Moe分析，能得到一组方程，另一台机器，被Joe分析，也能得到一组方程，两组方程一样。由于方程一样，所以，表现出来的现象也一样。于是，下面两个证据，就是一样的；一个证据用来证明：一台仪器放在新的地方的表现，与它在老地方的表现，是一样的；另一个证据用来证明，空间位置改变后，方程可以重复自己。因此，我们说，物理学的规律，对于位置转换来说，是对称的，对称的意义就是：当我们改变坐标系时，规律并不变化。当然，直观上显而易见，这是真的，然而，有趣且令人愉悦的则是，讨论其背后的数学。

11–3Rotations 11-3 旋转
The above is the first of a series of ever morecomplicated propositions concerning the symmetry of a physical law. The nextproposition is that it should make no difference in which direction wechoose the axes. In other words, if we build a piece of equipment in some placeand watch it operate, and nearby we build the same kind of apparatus but put itup on an angle, will it operate in the same way? Obviously it will not if it isa Grandfather clock, for example! If a pendulum clock stands upright, it worksfine, but if it is tilted the pendulum falls against the side of the case and nothinghappens. The theorem is then false in the case of the pendulum clock, unless weinclude the earth, which is pulling on the pendulum. Therefore we can make aprediction about pendulum clocks if we believe in the symmetry of physical lawfor rotation: something else is involved in the operation of a pendulum clockbesides the machinery of the clock, something outside it that we should lookfor. We may also predict that pendulum clocks will not work the same way whenlocated in different places relative to this mysterious source of asymmetry,perhaps the earth. Indeed, we know that a pendulum clock up in an artificial satellite,for example, would not tick either, because there is no effective force, and onMars it would go at a different rate. Pendulum clocks do involve somethingmore than just the machinery inside, they involve something on the outside.Once we recognize this factor, we see that we must turn the earth along withthe apparatus. Of course we do not have to worry about that, it is easy to do;one simply waits a moment or two and the earth turns; then the pendulum clockticks again in the new position the same as it did before. While we arerotating in space our angles are always changing, absolutely; this change doesnot seem to bother us very much, for in the new position we seem to be in thesame condition as in the old. This has a certain tendency to confuse one,because it is true that in the new turned position the laws are the same as in theunturned position, but it is not true that as we turn a thing itfollows the same laws as it does when we are not turning it. If we performsufficiently delicate experiments, we can tell that the earth is rotating,but not that it had rotated. In other words, we cannot locate itsangular position, but we can tell that it is changing.
还有很多更复杂的命题，牵扯到物理规律的对称性，上面就是其中之一。下一个命题则是，我们选择轴的方向是什么，应该没有什么不同。换句话说，如果我们把这个装备，在某地建立起来，并看着它运行，而在附近，我们建立同样的仪器，但是，让它转一个角度，它的运行，会一样吗？举个例子，如果它是老爷爷的钟，那么显然不行！如果一个单摆钟，站的笔直，那么，它会工作得很好，但是，如果它倾斜了，那么，单摆就会靠着一侧，什么就都不会发生了。因此，在单摆钟这种情况，那个定理就是错的，除非我们把地球也包括进来，它对单摆有吸引力。因此，如果对于旋转来说，我们相信物理规律的对称性：除了钟表的机制之外，还有其它的东西，卷入了单摆钟的运行，这种东西在它之外，我们应该去寻找。我们还可以预测，当单摆钟被放在不同的地方时，相对于这种神秘的对称源—或许是地球，单摆钟的工作，会不一样。确实，例如，我们知道，在人造地球卫星上，单摆钟也不会嘀嗒，因为没有有效的力，而在火星上，速率则会不同。单摆钟除了内部的机制外，确实还包含了更多的东西，它们包含着外面的一些东西。一旦我们认识到这个因素，我们就明白了，我们应该把地球与仪器一起转动。当然，我们无需担心这点，这很容易做到；简单地等待一会，地球就会转；然后，单摆钟就会在新的位置，开始嘀嗒，正如它以前所做那样。当我们在空间中旋转时，我们的角度总是在改变，这是绝对地；这种改变，似乎并不让我们感到很困惑，因为，在新的位置，我们的条件，与老的条件，似乎是一样的。这一点，似乎总容易迷惑人，因为，有一点是真的，即在转换后的新位置，规律与转换前的位置是一样的；但是，有一点并不为真，即当我们转了一个事物之后，它还遵循同样的规律，就像我们没有转它之前一样。如果我们执行足够精确的实验，我们可以告知，地球正在转动着，而不是它已经转动过了。换句话说，我们无法定位它的角度位置，但是，我们可以告知，它正在改变。

Fig. 11–2.Two coordinate systems havingdifferent angular orientations. 图11-2 两个坐标系，有不同的角度方位。
Now we may discuss the effects of angularorientation upon physical laws. Let us find out whether the same game with Joeand Moe works again. This time, to avoid needless complication, we shallsuppose that Joe and Moe use the same origin (we have already shown that the axescan be moved by translation to another place). Assume that Moe’s axes have rotatedrelative to Joe’s by an angle θ . The two coordinate systems are shown in Fig. 11–2, whichis restricted to two dimensions. Consider any point P having coordinates (x,y) in Joe’s system and (x′,y′) in Moe’s system. We shall begin, as in the previous case, byexpressing the coordinates x′ and y′ in terms of x , y , and θ . To do so, we first drop perpendiculars from P to all four axes and draw AB perpendicular to PQ .Inspection of the figure shows that x′ can be written as the sum of two lengths along the x′ -axis, and y′ as the difference of two lengths along AB . All these lengths are expressed in terms of x , y , and θ in equations (11.5),to which we have added an equation for the third dimension.
x′=xcosθ + y sinθ,
y′=ycosθ − xsinθ, (11.5)
z′=z
现在，我们可以讨论，角度方位对物理规律的影响。让我们看看，用Joe和 Moe来做这个游戏，是否可行。这次，为了避免不必要的复杂，我们将假设，Joe和Moe，用的是同一个原点（我们已经指出过，坐标轴可以通过转换，移到另一地方）。假设Moe的轴，相对于Joe的，转动了一个角度θ。图11-2所示的这两个坐标系，被限制在二维。考虑任一点P，在Joe的系统中，坐标为(x,y)，在Moe的系统中，坐标为 (x′,y′)。正如前一情况一样，我们将用x , y , 和 θ，来表示x′ 和 y′。要这样做，我们首先从点P，向四个轴做垂线，且做AB垂直于 PQ。从图中可看出，x′可被表示为沿着x′轴的两个长度之和， y′可被表示为沿着AB的两个长度之差。所有这些长度，都在方程组（11.5）中，用x , y , 和θ来表示了，我们还为它，增加了一个第三维的方程：
x′=xcosθ + y sinθ,
y′=ycosθ − xsinθ, (11.5)
z′=z

The next step is to analyze the relationship of forces as seen by thetwo observers, following the same general method as before. Let us assume thata force F , which has already been analyzed as having components Fxand Fy (as seen by Joe), is acting on a particle of mass m , located at point P in Fig. 11–2. Forsimplicity, let us move both sets of axes so that the origin is at P, as shown in Fig. 11–3. Moesees the components of F along his axes as Fx′ and Fy′ . Fx has components along both the x′ - and y′ -axes, and Fy likewise has components along both these axes. To express Fx′in terms of Fx and Fy , we sum these components along the x′ -axis, and in a like manner we can express Fy′in terms of Fx and Fy . The results are
Fx′=Fx cosθ + Fy sinθ,
Fy′=Fy cosθ − Fx sinθ, (11.6)
Fz′=Fz
下一步，与前面的步骤一样，就是分析两个观察者所看到的力之间的关系。我们假设，力F已经被分析为，具有分量Fx和Fy (由Joe所看到),且正作用于位于点P的质量 m ,见图 11–2。为了简单起见，我们移动两个轴，让原点在P，如图 11–3所示。 Moe看到，F沿着他的坐标轴的分量为Fx′ 和Fy′。 Fx在x′和y′ 轴上都有分量，Fy也是。要用Fx和 Fy来表示Fx′, 我们把它们沿着x′ 轴的分量加起来，同样，我们可以用Fx和 Fy来表示Fy′。结果就是：
Fx′=Fx cosθ + Fy sinθ,
Fy′=Fy cosθ − Fx sinθ, (11.6)
Fz′=Fz

It is interesting to note an accident of sorts, which is of extremeimportance: the formulas (11.5)and (11.6),for coordinates of P and components of F , respectively, are of identical form.
指出一个分类，非常有趣：公式(11.5)和 (11.6)分别是关于点P的坐标和 F的分量的，它们的形式相同。这点极为重要。

Fig. 11–3.Components of a force in the twosystems. 图11-3 一个力在两个坐标系中的分量。

As before, Newton’s laws are assumed to betrue in Joe’s system, and are expressed by equations (11.1).The question, again, is whether Moe can apply Newton’s laws—will the results becorrect for his system of rotated axes? In other words, if we assume that Eqs.(11.5)and (11.6)give the relationship of the measurements, is it true or not true that
m(d2x′m(d2y′m(d2z′/dt2)=Fx′,/dt2)=Fy′,/dt2)=Fz′? (11.7)
像以前一样，牛顿的规律，在Joe的坐标系中，被假定是真的，且通过公式（11.1）来表示。而问题则又是：Moe能应用牛顿规律吗？对于他的旋转了的坐标系，结果还是正确的吗？换句话说，如果我们假定公式(11.5)和(11.6)给出了测量的关系，那么，下面的是正确的吗？
(11.7)

To test these equations, we calculate the left and right sidesindependently, and compare the results. To calculate the left sides, wemultiply equations (11.5)by m , and differentiate twice with respect to time, assuming theangle θ to be constant. This gives
m(d2x′m(d2y′m(d2z′/dt2)=m(d2x/dt2)=m(d2y/dt2)=m(d2z/dt2)cosθ+m(d2y/dt2)cosθ−m(d2x/dt2)./dt2)sinθ,/dt2)sinθ, (11.8)
要验证这些公式，我们分别计算左边和右边，然后比较结果。要计算左边，我们用m乘以公式（11.5），对时间求两次微分，认为角度θ是常数。这给出：
(11.8)
We calculate the right sides of equations (11.7)by substituting equations (11.1)into equations (11.6).This gives
Fx′Fy′Fz′=m(d2x=m(d2y=m(d2z/dt2)cosθ+m(d2y/dt2)cosθ−m(d2x/dt2)./dt2)sinθ,/dt2)sinθ, (11.9)
我们计算公式(11.7)的右边，是通过把公式(11.1)，代入公式（11.6）。这给出：
(11.9)

11–4Vectors 11-4 矢量
Not only Newton’s laws, but also the other lawsof physics, so far as we know today, have the two properties which we callinvariance (or symmetry) under translation of axes and rotation of axes. Theseproperties are so important that a mathematical technique has been developed totake advantage of them in writing and using physical laws.
不仅牛顿规律，而且还有我们今天所知道的其他物理规律，在坐标轴的转换或坐标轴的旋转下，都有这两个属性，我们称之为不变性（或对称性）。这些属性是如此重要，以至于一种数学的技术，已经被发展出来，以利用它们的优势，来书写和使用物理规律。

The foregoing analysis involvedconsiderable tedious mathematical work. To reduce the details to a minimum inthe analysis of such questions, a very powerful mathematical machinery has beendevised. This system, called vector analysis, supplies the title of thischapter; strictly speaking, however, this is a chapter on the symmetry ofphysical laws. By the methods of the preceding analysis we were able to doeverything required for obtaining the results that we sought, but in practicewe should like to do things more easily and rapidly, so we employ the vectortechnique.
前面的分析，包含了相当冗长的数学工作。为了在这种问题的分析中，把细节降到最低，一个强有力的数学机制，被发明了出来。这个系统，被称为矢量分析，提供了这一章的标题，然而，严格说来，这一章是关于物理规律的对称的。通过前面分析中的方法，为了得到我们想要寻求的结果，我们可以做任何所要求的事情，但是，在实践中，我们喜欢以更简单和更快捷的方式来做事情，所以，我们雇佣了矢量技术。

We began by noting some characteristics oftwo kinds of quantities that are important in physics. (Actually there are morethan two, but let us start out with two.) One of them, like the number ofpotatoes in a sack, we call an ordinary quantity, or an undirected quantity, ora scalar. Temperature is an example of such a quantity. Other quantitiesthat are important in physics do have direction, for instance velocity: we haveto keep track of which way a body is going, not just its speed. Momentum andforce also have direction, as does displacement: when someone steps from one placeto another in space, we can keep track of how far he went, but if we wish alsoto know where he went, we have to specify a direction.
物理学中，有两类量，非常重要，我们通过指出其特性，开始讲。（实际上，多于两类，但让我们从两个开始。）其中之一，就像一袋土豆的数目一样，我们称之为：常规量、或没有指向的量、或标量。温度就是一个这种量。物理学中其他的重要的量，确实是有方向的，比如矢量速度：一个物体，往哪个方向走，我们必须记录其轨迹，而不仅仅是速度。动量和力，也有方向，正如位移一样：当一个人从一个地方，走向另一个地方，我们可以记录他走了多远，但是，如果我们也希望知道，他去过哪里，我们就必须具体指出方向。

All quantities that have a direction, likea step in space, are called vectors.
所有有一个方向的量，如像空间中的一步那样的，都被称为矢量。

A vector is three numbers. In order torepresent a step in space, say from the origin to some particular point Pwhose location is (x,y,z) , we really need three numbers, but we are going to invent a singlemathematical symbol, r , which is unlike any other mathematical symbols we have so far used.1It is not a single number, it represents three numbers: x , y , and z . It means three numbers, but not really only those threenumbers, because if we were to use a different coordinate system, the threenumbers would be changed to x′ , y′ , and z′ . However, we want to keep our mathematics simple and so we are goingto use the same mark to represent the three numbers (x,y,z)and the three numbers (x′,y′,z′) . That is, we use the same mark to represent the first set of three numbersfor one coordinate system, but the second set of three numbers if we are usingthe other coordinate system. This has the advantage that when we change thecoordinate system, we do not have to change the letters of our equations. If wewrite an equation in terms of x,y,z , and then use another system, we have to change to x′,y′,z′, but we shall just write r , with the convention that it represents (x,y,z) if we use one set of axes, or (x′,y′,z′) if we use another set of axes, and so on.如果我们用x,y,z The three numberswhich describe the quantity in a given coordinate system are called the componentsof the vector in the direction of the coordinate axes of that system. That is,we use the same symbol for the three letters that correspond to the sameobject, as seen from different axes. The very fact that we can say “thesame object” implies a physical intuition about the reality of a step in space,that is independent of the components in terms of which we measure it. So thesymbol r will represent the same thing no matter how we turn the axes.
一个矢量，是三个数字。为了表示空间中的一步，比如从原点到某个具体的点P，其位置为 (x,y,z)，我们确实需要三个数字，但是，我们将要发明一个单独的数学符号，r，一直以来，我们使用过很多其他的数学符号，但r与这些，都不相同。（脚注1）它不是一个单独的数字，它代表着三个数字：x , y , 和 z。它意味着三个数字，但并不仅仅是这三个数字，因为，如果我们使用不同的坐标系，则这三个数字就会变成x′ , y′ , 和 z′。然而，我们想让我们的数学，保持简单，所以，我们将使用此同一个标记，来表示这三个数字(x,y,z)和(x′,y′,z′)。也就是说，我们使用同一个标记，来表示第一个坐标系的第一组的三个数字，如果我们使用的是第二个坐标系，那么，它就表示的是第二组的三个数字。这样做的优势就是，当我们改变坐标系式，我们不用改变我们方程的字母。写一个方程，然后，使用另一个坐标系，我们就需要改成x′,y′,z′，但是，现在我们只写r，就很方便，因为，如果我们使用第一组坐标轴，它就代表着(x,y,z), 如果我们使用另一组坐标轴，它就代表着(x′,y′,z′)，如此等等。在一个坐标系中用来描述量的这三个数字，被称为矢量的分量；这个坐标系的坐标轴，是有方向的，矢量在其中，也是有方向的{？}。也就是说，同一个对象，从不同的坐标轴看，有不同的三个字母，我们使用同一个符号，与这三个字母相应。事实，我们能说“同一个对象”这一事实，就意味着，对于空间中的一步这一现实，我们有一个物理直观，我们测量这一现实，会用术语，来描述这些分量，该直观，独立于这些分量{术语}。于是，无论我们如何转动这个坐标轴，符号将始终代表着同一个事物。

Now suppose there is another directed physicalquantity, any other quantity, which also has three numbers associated with it,like force, and these three numbers change to three other numbers by a certainmathematical rule, if we change the axes. It must be the same rule that changes(x,y,z) into (x′,y′,z′) . In other words, any physical quantity associated with three numberswhich transform as do the components of a step in space is a vector. Anequation like
F=r
would thus be true in any coordinate system if it were true inone. This equation, of course, stands for the three equations
Fx=x, Fy=y, Fz=z,
or, alternatively, for
Fx′=x′, Fy′=y′, Fz′=z′.
The fact that a physical relationship can be expressed as a vectorequation assures us the relationship is unchanged by a mere rotation of thecoordinate system. That is the reason why vectors are so useful in physics.
现在，假设有另外一个直接的物理量，可以是任何量，也有三个与它关联的数字，比如力，如果我们改变坐标轴的话，这三个数字，就会依据某条数学规则，变为另外的三个数字。这条规则，与把(x,y,z)变成(x′,y′,z′)的规则，应该是同一条。换句话说，任何物理量，只要是与三个数字相关联的，且这三个数字，就像空间中的一步的分量一样，会转换，那么，该物理量就是一个矢量。一个像
F=r
的方程，如果在一个坐标系中是真的，那么，在任何坐标系中，也都会是真的，当然，这个方程，代表着三个方程：
Fx=x, Fy=y, Fz=z,
或者，可替代地为：
Fx′=x′, Fy′=y′, Fz′=z′.
物理关系，可被表示为矢量方程这一事实，保证了：仅仅旋转坐标系，这个关系不变。这就是为什么，在物理学中，矢量非常有用。